Find the minimum size square capable of bounding n equal squares arranged in any configuration. The first few cases are illustrated above (Friedman). The only packings which have been proven optimal are 2, 3, 5, 6, 7, 8, 14, 15, 24, and 35, in addition to the trivial cases of the square numbers (Friedman). If n = a^2 - a for some a, it is conjectured that the size of the minimum bounding square is a for small n. The smallest n for which the conjecture is known to be violated is n = 272 (with a = 17). The following table gives the smallest known side lengths for a square into which n unit squares can be packed. An asterisk (*)indicates that a packing has been proven to be optimal.

circle packing | packing | square dissection | triangle packing